This is a feature that will soon be implemented in Alice 3, but we are still working on it at this time...

At this time I would recommend using screen capture software, such as Camtasia or Jing (which is open-source and free to download and install: http://www.techsmith.com/jing.html), to capture the playback of the Alice animation on your computer screen.

Some of these, including Jing, are 30-day trials, followed by a pay-for-me after you get comfortable with it.
Also, be careful, and watch the install closely, as some will also ask for permission to add Bloatware during the process.

For Windows users: I've been using Windows Media Encoder 9 Series x64 edition with Win7 x64. It's also supposed to work with Win8.1. Haven't found any problems with it yet, and it's also free.